I'll answer this (good) question directly after some preliminaries. (It was posed ~10 years ago but other questions, more recent, are substantively similar.)
But I think it's important to make a comment on the selected answer. This answer (by Adam) is misleading at best. I'd like to say it in a more kind way but, honestly, it's wrong.
As an example, large sections of scattering and reaction theory texts are devoted to "rearrangement" processes. These are processes* where the participating nuclei, for example, in the initial and final states are all different nuclei. This means that the individual nucleons are transferred between the particles in the initial state to give another set of partitions (corresponding to different nuclei) in the final state. A complete explanation of how the LS describes many-body scattering and reaction theory can be found in the following texts: Weinberg, QTFv1, Ch3-4 (et seq.); Newton, Scattering theory of waves and particles; Goldberger & Watson, Collision Theory.
It might be worth mentioning that, in principle, the physics content of LS and SD equations are the same. (I say "in principle" because I don't believe this has been rigorously proven -- please correct me if I'm wrong about this. But neither am I aware if it has been disproved.) The use of SD equations became necessary when it was realized (sometime in the early- to mid-60's by people like Gell-Mann, Goldberger, Weinberg, Low, etc.) that the requirement of Lorentz invariance and other, internal symmetries that must be observed, are difficult to implement in the LS approach. But these are straightforward in the relativistically invariant approaches that begin with the canonical formalism. (See Weinberg, QTFv1, Ch.7.)
Additionally, it's wrong to say that the LS equation cannot handle reactions where the numbers (and types) of particles change. If the Hamiltonian has interaction terms that allow the number of particles to vary -- consider, for example, the pion-nucleon interaction $\mathscr{H}_{\pi N} = g_{\pi N} \phi^i_\pi(x) \bar\psi(x) \tau_i \gamma_5 \psi(x)$ -- then the LS equation will predict a non-zero amplitude for the process $\pi^0 n \to \pi^+ \pi^- \pi^- p$, as one example.
In order to answer the OP's questions, we define
$$G = [E-H]^{-1}$$
for the Lippmann-Schwinger equation and for what the OP called "$G$" in the SD approach, we will have $G\to\Delta$ (in a highly schematic form):
$$\Delta = \frac{\delta^2}{\delta J \delta J}W[J],$$
where $W$ is the logarithm of the $n$-point correlator generating functional $Z$. $\delta/\delta J$ is a functional derivative and the two functional derivatives means that $\Delta$ is the two-point function. We also define
$$\Pi = \frac{\delta^2}{\delta \Phi \delta \Phi}\Gamma[\Phi],$$
where $\Gamma$ is the quantum effective action. The relationship between these two quantities is $$\Delta = \Pi^{-1}$$ (again, schematically, in a sort of matrix notation). Defining $\Delta_0, \Pi_0$ for the non-interacting theory $Z\to Z_0, \Gamma\to\Gamma_0$ with $\Delta_0 = \Pi_0^{-1}$, and defining $\Pi = \Pi_0 - \Pi^*$, gives:
$$\Delta = \Delta_0 + \Delta_0 \Pi^* \Delta.$$ Note that the OP's equation for SD, $G = G_0 + G_0 \Sigma G$ is recovered if we allow $\Delta\to G$, $\Delta_0\to G_0$ and $\Pi^*\to\Sigma$. Also, note the similarity of the structure of this equation to that of the OP's LS equation $G=G_0 + G_0 V G$.
We can get a sense of the physical contents of the respective approaches -- LS & SD -- by giving the explicit forms for the LS & SD propagators -- the free particle terms $G_0$ and $\Delta_0$ as:
$$(\Phi_k, G_0(E) \Phi_{k'}) = \frac{1}{E+i\epsilon-E_k} (\Phi_k,\Phi_{k'}),$$
where $\Phi_k$ is the free-particle state vector of $N$ particles. On the other hand, the SD free-particle Green function (aka, "propagator") is:
$$\Delta_0 = (p^2 - (m^2-i\epsilon))^{-1}.$$
My questions are
Are the two Green's functions the same?
No. They satisfy similar-looking equations. But their meaning is different. (We could analyze the pole structures of $G_0$ and $\Delta_0$ to prove this.) They carry -- in a sense -- different parts of the same physics.
What's the relation between the two formalisms?
The LS equation is derived starting from the Heisenberg representation of (many-body, we might emphasize) relativistic field theory. The SD equation is derived from the canonical formalism where a Lagrangian with the correct symmetries is used to derive the appropriate Hamiltonian. (See Weinberg, op cit., Ch.7, again.) To the extent that these formalisms are both expressions of relativistic quantum field theory, they should give the same predictions for measurements that can be computed in both approaches.
And the relation between Lippmann-Schwinger equation and Dyson equation?
The have a similar form but, as mentioned, carry different parts of the physics.
If they are actually the same thing, then does it mean =Σ(this sounds very stupid)?
Not a stupid question. But, the answer is no. In other words, $V\ne \Sigma$, generally.
Are the possible differences relating to the discrepancy between S-matrix theory and QFT?
It's not what you're asking but I want to emphasize, as alluded to before (see the answer to "What's the relation between the two formalisms?"), these two approaches (LS & SD) should give the same results when observables, which are the exclusive province of the $S$ matrix, can be calculated either way. You're now asking, however, about S-matrix theory. (For what I mean by this term, see the book The Analytic S-Matrix by Eden, Landshoff, Olive, & Polkinghorne.) To the extent that the program of the $S$-matrix theory could, in principle, be carried out it is completely consistent with relativistic quantum field theory (at least, in flat space). In fact, the problem with $S$-matrix theory is that it is vacuous with respect to the dynamics of interacting systems -- it's tenets are satisfied by all (effective) relativistic quantum field theories. So for $S$-matrix theory to be useful it must pull itself up by it's own "bootstraps" (this is "the bootstrap" or "bootstrapping") either from experimentally observed data or from some assumed model field theory.
*We could make similar statements, without essential modifications, for scattering and reactions where the particles are not just nuclei but baryons and mesons, such as $\pi^+ n \to \eta p.$
